Knowlpedia

Identity

Angular-velocity equation in axisymmetric flow

The swirl equation after dividing azimuthal velocity by radius, with the radial heat operator of four transverse dimensions.

Core idea

For a smooth axisymmetric solution with r>0r>0, write uθ=rΩu_\theta=r\Omega. The azimuthal equation is equivalent to

(∂t+ur∂r+uz∂z)Ω+2urrΩ=ν(∂r2+3r∂r+∂z2)Ω+fθr.\left(\partial_t+u_r\partial_r+u_z\partial_z\right)\Omega +\frac{2u_r}{r}\Omega =\nu\left(\partial_r^2+\frac3r\partial_r+\partial_z^2\right)\Omega +\frac{f_\theta}{r}.
Derivation

The gives D(rΩ)=rDΩ+urΩ\mathcal D(r\Omega)=r\mathcal D\Omega+u_r\Omega. The additional curvature term uruθ/ru_ru_\theta/r gives another urΩu_r\Omega. On the diffusion side,

(Δ0−r−2)(rΩ)=r(∂r2+3r∂r+∂z2)Ω.(\Delta_0-r^{-2})(r\Omega) =r\left(\partial_r^2+\frac3r\partial_r+\partial_z^2\right)\Omega.

Dividing the resulting equation by rr proves the formula.

Radial interpretation

The operator ∂r2+3r−1∂r\partial_r^2+3r^{-1}\partial_r is the radial Laplacian in four Euclidean dimensions. This algebraic reformulation is useful for heat-profile constructions; it does not change the physical fluid's three-dimensional domain.